Suppose I give you a prime $p$ and ask for a non-CM supersingular elliptic curve over $\mathbb{F}_p.$ Can this be done in polynomial time (so, polynomial in $\log p$)?
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asked Nov 16, 2012 at 18:54
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$\begingroup$ I'm confused. All elliptic curves over $\mathbb{F}_p$ have CM (if CM means endormorphism ring not equal to $\mathbb{Z}$). $\endgroup$– David E SpeyerCommented Nov 16, 2012 at 19:21
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$\begingroup$ What's the difference between a CM and non-CM supersingular elliptic curve over a finite field? $\endgroup$– Will SawinCommented Nov 16, 2012 at 19:26
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$\begingroup$ I mean, not a reduction of a CM curve over $\mathbb{Q}.$ Sorry about the shaky terminology, I am just learning about all this. $\endgroup$– Igor RivinCommented Nov 16, 2012 at 19:58
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$\begingroup$ Over $\mathbb{F}_{11}$ (or the algebraic closure), there are only two supersingular elliptic curves up to isomorphism, one is the reduction of an elliptic curve with j invariant zero and one is the reduction of an elliptic curve with j-invariant 1728. So you must mean something different. $\endgroup$– stankewiczCommented Nov 16, 2012 at 21:24
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2$\begingroup$ Igor: every elliptic curve over a finite field is the reduction of a CM elliptic curve over the corresponding unframified extension of Q_p. This is part of the Deuring Lifting Theorem. When the curve is ordinary (= not supersingular) one can choose the lift so that the reduction map on the endomorphism rings is an isomorphism: this is called the canonical lift. Your question seems to need some revision... $\endgroup$– Pete L. ClarkCommented Nov 17, 2012 at 0:47
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